Compare rectangles with different width to height ratio.
No, see solution.
Practice makes perfect
Our friend says that all rectangles with the same perimeter as Triangle QRS will automatically have the same area as the triangle. To disprove this claim, we must find one rectangle whose perimeter equals that of Triangle QRS, and whose area does not equal that of Triangle QRS. To begin, let's calculate the perimeter and area of the triangle.
Triangle QRS
Let's let the base of QRS be RS and the height be QR. To find the length of QS, we can calculate the distance between Q and S using the Distance Formula.
Thus, the length of QS is 5. To calculate the triangle's perimeter, we can add the lengths of all three sides. This gives
P = 3+4+5 = 12.
To calculate the area of the triangle, we can use
A=1/2bh.
Let's substitute b with RS=4 and h with QR=3. This gives
A=1/2bh=1/2* 4 * 3=6.
Thus, for Triangle QRS, we have
P=12 and A=6.
Rectangle ABCD
Let's call our rectangle Rectangle ABCD. We want to create a rectangle such that its perimeter is 12 and its area is not 6. The perimeter of a rectangle can be found using the formula
P = 2l+2w
where l is the length of the rectangle and w its width. If our friend is correct, every rectangle with the perimeter 12 will have the area 6. We can create Rectangle ABCD such that its perimeter equals 12. Let's arbitrarily decide to make the width 2. We can now calculate the length the rectangle will have.
If Rectangle ABCD has a width of 2 and a length of 4, its perimeter is 12, the same as Triangle QRS. Let's now calculate the area of ABCD using the formula A=l w.
A = l w = 4* 2 = 8
The area of our rectangle is 8, while the area of the triangle is 6. Thus, our friend is incorrect. A rectangle with the same perimeter as a triangle will not always have the same area.
